Solve ((4×8)^-5)^4: Nested Exponents with Negative Powers

To solve this problem, we need to simplify the expression $\left(\left(4 \times 8\right)^{-5}\right)^4$.

We'll follow these steps:

• Step 1: Understand the expression inside the parentheses $(4 \times 8)$ and calculate it.

• Step 2: Apply the power of a power property to simplify the expression.

Step 1: Calculate the expression inside the parentheses.
We have $4 \times 8 = 32$, so the expression becomes $\left(32^{-5}\right)^4$.

Step 2: Apply the power of a power property.
This property states that $(a^m)^n = a^{m \cdot n}$. Here, $m = -5$ and $n = 4$, so:

$(32^{-5})^4 = 32^{-5 \times 4} = 32^{-20}$.

Therefore, the simplified expression is $\left(4 \times 8\right)^{-20}$.

Hence, the correct answer choice is:

• Choice 4: $\left(4\times8\right)^{-20}$

All other choices result from errors in applying the exponent rules or miscalculating intermediate steps:

• Choice 1: Misapplies the exponent rules, yielding $-9$ instead of $-20$.

• Choice 2: Incorrectly calculates the expression, resulting in $-1$.

• Choice 3: Incorrect fractional exponent interpretation does not apply here.